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Re: Globally limiting precision or accuracy
*To*: mathgroup at smc.vnet.net
*Subject*: [mg61032] Re: [mg61010] Globally limiting precision or accuracy
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Sat, 8 Oct 2005 02:48:40 -0400 (EDT)
*References*: <200510070737.DAA03251@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
On 7 Oct 2005, at 16:37, Lee Newman wrote:
> Dear Group,
>
> Situation
> - I have a large neural network model (my own code, not Wolfram's
> toolbox) that currently has a run time of about 20hours.
> - I am in the process of trying to profile and optimize the code
> (mostly matrix computations) to reduce the run time.
> - All of the computationas that I do are numerical.
>
> Questions
> (1) If I am not concerned about numerical accuracy beyond 3 decimal
> places for any of the computations in the model, can I improve
> performance by telling mathematica to globally restrict its
> accuracy (or
> precision) for all computations?
>
> (2) If so, how do I do this? Is it as simple as setting
> $MachinePrecision=3?
I don't think this can be done at all. You can of course Unprotect
[$MachinePrecision] and set the global $MachinePrecision variable to
3, but this won't make any difference at all to machine precision
calculations because (as the name tells you) that is decided by the
"machine" i.e. your processor.
I am pretty sure there is no way to improve on the performance of
machine precision floating point computations by any Mathematica
commands. In fact, I think, the only way to do that is to get a
faster computer.
On the other hand you should be aware that when you use machine
precision arithmetic Mathematica does not attempt to keep any track
of the precision of your computations. If you come across an ill-
conditioned expression it will return nonsensical answers without any
warning. You certainly cannot be sure that any machine precision
computation is accurate even to three digits. To be sure of that you
have to work with extended precision numbers, because only then
Mathematica will keep track of precision. By using exact quantities
as input in your expression and computing N[expr,3] you should
normally get guaranteed three digits of precision in a relatively
efficient way. E.g.
N[Pi+23/234,3]
3.24
Precision[%]
3.
This will always be slower than when you do this with
MachinePrecision (which formally gives more digits of "precision" but
without any guarantee.
> Is there a global way (rather than local use of
> N[]) to ensure that all computations are done numerically, and with
> machine precision?
>
As long as the numbers in your input are machine precision numbers,
e.g. 1.3 or 2.3456 etc, all computations will be done with machine
precision, with all that it entails (including the possibility of
nonsensical answers). In fact, usually it is enough that you
expression involves a single machine precision number for the entire
expression to be computed using machine precision as in
1.2*Pi+123
126.77
Precision[%]
MachinePrecision
Andrzej Kozlowski
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